Asymptotically optimal assignments in ordinal evaluations of proposals

In ordinal evaluations of proposals in peer review systems, a set of proposals is assigned to a fixed set of referees so as to maximize the number of pairwise comparisons of proposals under certain referee capacity and proposal subject constraints. The following two related problems are considered: (1) Assuming that each referee has a capacity to review k out of n proposals, 2 ≤ k ≤ n, determine the minimum number of referees needed to ensure that each pair of proposals is reviewed by at least one referee, (2) Find an assignment that meets the lower bound determined in (1). It is easy to see that one referee is both necessary and sufficient when k = n, and n(n-1)/2 referees are both necessary and sufficient when k = 2. It is shown that 6 referees are both necessary and sufficient when k = n/2. Furthermore it is shown that 11 referees are necessary and 12 are sufficient when k = n/3, and 18 referees are necessary and 20 referees are sufficient when k = n/4. A more general lower bound of n(n-1)/k(k-1) referees is also given for any k, 2 ≤ k ≤ n, and an assignment asymptotically matching this lower bound within a factor of 2 is presented. These results are not only theoretically interesting but they also provide practical methods for efficient assignments of proposals to referees.